# Topology

## Topology (Encyclopedia of Science)

Topology is a branch of mathematics sometimes known as rubber-sheet geometry. It deals with the properties of a geometric figure that do not change when the shape is twisted, stretched, or squeezed. In topological studies, the tearing, cutting, and combining of shapes is not allowed. The geometric figure must stay intact while being studied. Topology has been used to solve problems concerning the number of colors necessary to illustrate maps, about distinguishing the characteristics of knots, and about understanding the structure and behavior of DNA (deoxyribonucleic acid) molecules, which are responsible for the transferring of physical characteristics from parents to offspring.

Topological equivalency

The crucial problem in topology is deciding when two shapes are equivalent. The term equivalent has a somewhat different meaning in topology than in Euclidean geometry. In Euclidean geometry, one is concerned with the measurement of distances and angles. It is, therefore, a form of quantitative analysis. In contrast, topology is concerned with similarities in shape and continuity between two figures. As a result, it is a form of qualitative analysis.

For example, in Figure 1 on page 1898, each of the two shapes has five points: a through e. The sequence of the points does not change from shape 1 to shape 2, even though the distance between the...

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## Topology (International Dictionary of Psychoanalysis)

Topology refers primarily to the branch of mathematics that rigorously treats questions of neighborhoods, limits, and continuity. Psychoanalysts have applied it to the study of unconscious structures.

In what have been called his two "topographies" (the first dating from 1900 and the second from 1923), Freud resorted to schemas to represent the various parts of the psychic apparatus and their interrelations. These schemas implicitly posited an equivalence between psychic and Euclidean space.

Early on, Jacques Lacan noted that the limitations of such a naive topology had restricted Freudian theory, not only in the description of the psychic apparatus (a description that in the end required an appeal to the economic point of view), but also in the specificity of clinical structures. The hypothesis that the unconscious is structured like a language, that is, in two dimensions, led Lacan to the topology of surfaces. The concept of foreclosure, for example, which he constructed on the basis of this topology, confirmed the heuristic value of his approach.

In his seminar "Identification" (1961-1962), Lacan unveiled a collection of topological objectsuch as the torus, the Möbius strip, and the cross-caphat served pedagogical aims. But already he saw them as more than just models. With the Borromean knot, introduced in 1973, he took the position that these objects were a real presentation of the subject and not just a representation. Below are several of Lacan's topological objects.

1. The Cut and the Signifier

Far from being given a priori, every space is organized on the basis of cuts and can actually be considered as a cut in the space of a higher dimension. We are familiar with the subjective impact of this: The events of our lives only become history through the castration complex, which organizes our reality at the price of an imaginary cutting off of the penis. According to Freud, by introjecting a single trait of another, the subject identifies with the other (at the price of losing this person as a love object). In the single trait Lacan found the very structure of the signifier: A cut allows the lost object to fall away. He called this cut the "unary trait."

The linguist Ferdinand de Saussure insisted on the fundamentally negative, purely differential character of the signifier. Lacan formalized this property in the double loop, or "interior eight," in which the gap created by the cut is closed after a second trip around a fictional axis. The difference of the signifier from itself is indicated by the difference between the two trips around the loop (Figure 1).

2. The Möbius Strip and Interpretation

If a signifier represents the subject for another signifier, then the subject would be supported by a surface whose edge would be a signifying cut. Note that the planehe usual screen for the subject's images, figures, and dreams, that is, planss a surface that does not meet these conditions. The double loop cannot be drawn on a plane without showing a cut. The same is true of a sphere, a simple representation of the universe.

The Möbius strip, on the other hand, can represent this cut and symbolize the subject of the unconscious. Since a Möbius strip only has one surface, it is possible to pass from one side to the other without crossing over any edgen apt representation of the return of the repressed. The Möbius strip also has certain other peculiarities. A cut that runs one-third from the edge and parallel to the edge divides the strip into a two-sided strip linked to what remains of the original Möbius strip. But if this cut is made in the center, it does not divide the Möbius strip in two. Instead, the entire strip is transformed into a strip with two sides. This characteristic illustrates the equivalence between the Möbius strip (the subject) and the medial cut that transforms it, and also provides a model of how interpretation functions. Interpretation does not abolish the unconscious. On the contrary, it makes the unconscious real for the subject by its transformed appearance as another (an Other) surface (figure 2).

3. The Torus

Lacan made different uses of the torus. By drawing Venn diagrams, traditionally used to illustrate basic logical operations, on the surface of the torus, he demonstrated the extent to which our thinking depends upon the plane surface, and he also provided another possible basis for the logic of the unconscious (Figure 3).

By inscribing the same circles on the surface of the torus, Lacan revealed the logic of the unconscious discovered by Freud (Figure 4).

On the torus, only symmetrical difference is consistent. Thus we have a demonstration of how the signifier can be different from all other signifiers and also from itself.

Lacan also used the torus to represent the subject as the subject of demand. In this sense, the torus can be conceived as the surface created by the iteration of the trajectory of the subject's demand. This trajectory turns around two different empty spaces, one that is "internal," D, the lack created in the real by speech, and one that is "central," d, corresponding to the place of the elusive object of desire that the drive goes around before completing the loop (Figure 5).

For every torus, there is a complementary torus, and the empty spaces of the two are the inverse of each other. Lacan made this structure of complementary toruses the support of the neurotic illusion that makes the demand of the Other the object of subject's desire and, conversely, makes the desire of the Other the object of subject's demand. This structure also arises from the fact that on a torus, the signifying cut (the double loop) does not detach any fragment. Neurotic subjects, insofar as they give in to neurosis, insofar as they are "in the torus," are not organized around their own castration, but instead excuse themselves by substituting the Other's demand for the object of their fantasy (figure 6).

4. The Cross-Cap

The cross-cap, or more precisely, the projective plane, can represent the subject of desire in relation to the lost object. A double loop drawn on its surface in effect divides this single-sided surface into two heterogeneous parts: a Möbius strip representing the subject and a disk representing object a, the cause of desire. The disk is centered on a point that is related to the irreducible singularity of this surface, which Lacan identified with the phallus. Unlike the representation of the subject produced on the torus, here a single cut, which symbolizes castration, produces both the subject and the object in its divisions (figure 7).

5. The Borromean Knot

Introduced by Lacan in 1973, the Borromean knot is the solution to a problem perceivable only in Lacanian theory but having extremely practical clinical applications. The problem is: How are the three registers posited as making up subjectivityhe real (R), the symbolic (S), and the imaginary (I)eld together?

Indeed, the symbolic (the signifier) and the imaginary (meaning) seem to have hardly anything in common fact demonstrated by the abundance and heterogeneity of languages. Moreover, the real, by definition, escapes the symbolic and the imaginary, since its resistance to them is precisely what makes it real.

This is why Lacan identified the real with the impossible.) In psychoanalysis, the real resists, and thus is distinct from, the imaginary defenses that the ego uses specifically to misrecognize the impossible and its consequences.

If each of the three registers R, S, and I that make up the Borromean knot is recognized to be toric in structure and the knot is constructed in three-dimensional space, it constitutes the perfect answer to the problem above, because it realizes a three-way joining of all three toruses, while none of them is actually linked to any other: If any one of them is cut, the other two are set free. Reciprocally, any knot that meets these conditions is called Borromean. Note that the subject is now defined by such a knot and not merely, as with the cross-cap, as the effect of a cut (figure 8).

Unfortunately, this ideal solution, which could be considered normal (without symptoms), seems to lead to paranoia. Lacan considered this to be the result of failure to distinguish among the three registers, as if they were continuous, which indeed occurs in clinical work. Being identical, R, S, and I are only differentiated by means of a "complication," a fourth ring that Lacan called the "sinthome." By making a ring with the three others, the sinthome (symptom) differentiates the three others by assuring their knotting (figure 9).

In this arrangement, the sinthome has the function of determining one of the rings. If it is attached to the symbolic, it plays the role of the paternal metaphor and its corollary, a neurotic symptom.

Lacan also drew upon non-Borromean knots, generated by "slips," or mistakes, in tying the knots. These allowed him to represent the status of subjects who are unattached to the imaginary or the real and who compensate for this with supplements (Lacan, 2001). In such cases the sinthome is maintained.

By using knots, Lacan was able to reveal his ongoing research without hiding its uncertainties. The value of the knots, which resist imaginary representation, is that they advance research that is not mere speculation and that they can graspt the cost of abandoning a grand synthesis few "bits of the real" (Lacan, 1976-1977, session of March 16, 1976). Even though he knew something about topology as practiced by mathematicians, Lacan advised his students "to use it stupidly" (Lacan, 1974-1975, session of December 17, 1974) as a remedy for our imaginary simplemindedness. He also recommended manually working with the knots by cutting surfaces and tying knots. Finally, for Lacan, topology had not only heuristic value but also valuable implications for psychoanalytic practice.

BERNARD VANDERMERSCH

See also: Knot; L and R schemas; Seminar, Lacan's; Signifier/signified; Structural theories; Symptom/sinthome; Thalassa. ATheory of Genitality; Unary trait.

Bibliography

Bourbaki, Nicolas. (1994). Elements of the history of mathematics (John Meldrum, Trans.). Berlin: Springer-Verlag.

Darmon, Marc. (1990). Essais sur la topologie Lacanienne. Paris:itions de l'Association Freudienne Internationale.

Lacan, Jacques. (1975). La troisième, intervention de J. Lacan, le 31 octobre 1974. Lettres de l 'ole Freudienne, 16, 178-203.

. (1974-1975). Le séminaire, livre XXII, R.S.I. Ornicar? 2-5.

. (1976-1977). Le séminaire XXIII, 1975-76: Le sinthome. Ornicar? 6-11.

. (2001). Joyce: Le symptôme. In his Autres écrits. Paris: Seuil.

Pont, Jean-Claude. (1974). La topologie algébrique des origines à Poincaré. Paris: Presses Universitaires de France.